In
mathematics, Khovanov homology is an oriented
link invariant that arises as the
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
of a
cochain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of ...
. It may be regarded as a
categorification
In mathematics, categorification is the process of replacing set-theoretic theorems with category-theoretic analogues. Categorification, when done successfully, replaces sets with categories, functions with functors, and equations with natural ...
of the
Jones polynomial
In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomia ...
.
It was developed in the late 1990s by
Mikhail Khovanov
Mikhail Khovanov (russian: Михаил Гелиевич Хованов; born 1972) is a Russian- American professor of mathematics at Columbia University who works on representation theory, knot theory, and algebraic topology. He is known for in ...
, then at the
University of California, Davis
The University of California, Davis (UC Davis, UCD, or Davis) is a public land-grant research university near Davis, California. Named a Public Ivy, it is the northernmost of the ten campuses of the University of California system. The institu ...
, now at
Columbia University.
Overview
To any link diagram ''D'' representing a
link ''L'', we assign the Khovanov bracket
''D''">/nowiki>''D''/nowiki>, a cochain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of ...
of graded vector space
In mathematics, a graded vector space is a vector space that has the extra structure of a '' grading'' or a ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces.
Integer gradation
Let \mathbb be th ...
s. This is the analogue of the Kauffman bracket in the construction of the Jones polynomial
In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomia ...
. Next, we normalise ''D''">/nowiki>''D''/nowiki> by a series of degree shifts (in the graded vector space
In mathematics, a graded vector space is a vector space that has the extra structure of a '' grading'' or a ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces.
Integer gradation
Let \mathbb be th ...
s) and height shifts (in the cochain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of ...
) to obtain a new cochain complex C(''D''). The cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
of this cochain complex turns out to be an invariant
Invariant and invariance may refer to:
Computer science
* Invariant (computer science), an expression whose value doesn't change during program execution
** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
of ''L'', and its graded Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space' ...
is the Jones polynomial of ''L''.
Definition
This definition follows the formalism given in Dror Bar-Natan's 2002 paper.
Let denote the ''degree shift'' operation on graded vector spaces—that is, the homogeneous component in dimension ''m'' is shifted up to dimension ''m'' + ''l''.
Similarly, let 's''denote the ''height shift'' operation on cochain complexes—that is, the ''r''th vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but ca ...
or module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
in the complex is shifted along to the (''r'' + ''s'')th place, with all the differential maps being shifted accordingly.
Let ''V'' be a graded vector space with one generator ''q'' of degree 1, and one generator ''q''−1 of degree −1.
Now take an arbitrary diagram ''D'' representing a link ''L''. The axioms for the Khovanov bracket are as follows:
# 'ø'''' = 0 → Z → 0, where ø denotes the empty link.
# ''O ''D'''' = ''V'' ⊗ 'D'''', where O denotes an unlinked trivial component.
# 'D'''' = F(0 → 0">'D''0'' → 1">'D''1'' → 0)
In the third of these, F denotes the `flattening' operation, where a single complex is formed from a double complex by taking direct sums along the diagonals. Also, ''D''0 denotes the `0-smoothing' of a chosen crossing in ''D'', and ''D''1 denotes the `1-smoothing', analogously to the skein relation
Skein relations are a mathematical tool used to study knots. A central question in the mathematical theory of knots is whether two knot diagrams represent the same knot. One way to answer the question is using knot polynomials, which are invaria ...
for the Kauffman bracket.
Next, we construct the `normalised' complex C(''D'') = 'D'''' −">��''n''− where ''n''− denotes the number of left-handed crossings in the chosen diagram for ''D'', and ''n''+ the number of right-handed crossings.
The Khovanov homology of ''L'' is then defined as the cohomology H(''L'') of this complex C(''D''). It turns out that the Khovanov homology is indeed an invariant of ''L'', and does not depend on the choice of diagram. The graded Euler characteristic of H(''L'') turns out to be the Jones polynomial of ''L''. However, H(''L'') has been shown to contain more information about ''L'' than the Jones polynomial
In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomia ...
, but the exact details are not yet fully understood.
In 2006 Dror Bar-Natan developed a computer program to calculate the Khovanov homology (or category) for any knot.
Related theories
One of the most interesting aspects of Khovanov's homology is that its exact sequences are formally similar to those arising in the Floer homology
In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer int ...
of 3-manifolds
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...
. Moreover, it has been used to produce another proof of a result first demonstrated using gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups ...
and its cousins: Jacob Rasmussen's new proof of a theorem of Peter Kronheimer
Peter Benedict Kronheimer (born 1963) is a British mathematician, known for his work on gauge theory and its applications to 3- and 4-dimensional topology. He is William Caspar Graustein Professor of Mathematics at Harvard University and former ...
and Tomasz Mrowka, formerly known as the Milnor conjecture
In mathematics, the Milnor conjecture was a proposal by of a description of the Milnor K-theory (mod 2) of a general field ''F'' with characteristic different from 2, by means of the Galois (or equivalently étale) cohomology of ''F'' wi ...
(see below). There is a spectral sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they hav ...
relating Khovanov homology with the knot Floer homology of Peter Ozsváth
Peter Steven Ozsváth (born October 20, 1967) is a professor of mathematics at Princeton University. He created, along with Zoltán Szabó, Heegaard Floer homology, a homology theory for 3-manifolds.
Education
Ozsváth received his Ph.D. from P ...
and Zoltán Szabó (Dowlin 2018). This spectral sequence settled an earlier conjecture on the relationship between the two theories (Dunfield et al. 2005). Another spectral sequence (Ozsváth-Szabó 2005) relates a variant of Khovanov homology with the Heegaard Floer homology of the branched double cover along a knot. A third (Bloom 2009) converges to a variant of the monopole Floer homology of the branched double cover. In 2010 Kronheimer and Mrowka exhibited a spectral sequence abutting to their instanton knot Floer homology group and used this to show that Khovanov Homology (like the instanton knot Floer homology) detects the unknot.
Khovanov homology is related to the representation theory of the Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
sl2. Mikhail Khovanov and Lev Rozansky have since defined homology theories associated to sl''n'' for all ''n''. In 2003, Catharina Stroppel extended Khovanov homology to an invariant of tangles (a categorified version of Reshetikhin-Turaev invariants) which also generalizes to sl''n'' for all ''n''.
Paul Seidel and Ivan Smith have constructed a singly graded knot homology theory using Lagrangian intersection Floer homology
In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer int ...
, which they conjecture to be isomorphic to a singly graded version of Khovanov homology. Ciprian Manolescu has since simplified their construction and shown how to recover the Jones polynomial from the cochain complex underlying his version of the Seidel-Smith invariant.
The relation to link (knot) polynomials
At International Congress of Mathematicians
The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU).
The Fields Medals, the Nevanlinna Prize (to be renam ...
in 2006 Mikhail Khovanov provided the following explanation for the relation to knot polynomials from the view point of Khovanov homology. The skein relation
Skein relations are a mathematical tool used to study knots. A central question in the mathematical theory of knots is whether two knot diagrams represent the same knot. One way to answer the question is using knot polynomials, which are invaria ...
for three links and is described as
:
Substituting leads to a link polynomial invariant